El proyecto de investigación “Numerical Analysis of Matrix Functions”, desarrollado durante los años 2001 y 2004, y auspiciado por la agencia de investigación “Engineering and Physical Sciences Research Council” (EPSRC), ha permitido un notable desarrollo teórico y práctico del cálculo de funciones de matrices:
• Método general para el cálculo de funciones de matrices complejas ([DaH103]). • Métodos para calcular funciones trascendentes y raíces p-ésimas.
• Técnicas que permiten obtener f(A)b, siendo f(z) función analítica, A matriz cuadrada y b vector, sin calcular explícitamente f(A) ([DaH203]).
• Teoría de la perturbación y números de condición asociados al cálculo de funciones de matrices ([Davi04]).
• Teoría y métodos para las funciones matriciales no primarias ([HiMN03], [HiMN03], [DaSm02]).
El software disponible (http://www.maths.man.ac.uk/~higham/NAMF/), consiste en un conjunto de ficheros de tipo “M” y de tipo “MEX” de Matlab, para el cálculo de funciones de matrices basados en el Algoritmo 1 del apartado 3.7.
5 Conclusiones
En este informe técnico se han descrito los algoritmos más relevantes para el cálculo de las funciones matriciales más usuales. Posiblemente pueden existir más métodos, pero no tienen la importancia de los descritos en este informe. No se puede asegurar cual es el mejor método, pues la eficiencia y precisión de cualquiera de ellos depende de la matriz y de la función considerada. Como se puede fácilmente observar, los métodos son muy dependientes de la función considerada, aunque emergen dos métodos que se aplican en todos los casos y que en general suelen presentar buenas cualidades numéricas: métodos basados en la forma real de Schur y los métodos basados en aproximaciones racionales de Padé con escalado. Una continuación del presente informe técnico consistirá en diseñar algoritmos generales basados en estos métodos que permitan un cálculo eficiente y preciso de funciones de matrices.
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